# Operator inference of non-Markovian terms for learning reduced models from partially observed state trajectories

arXiv: https://arxiv.org/abs/2103.01362 · HTML: https://mlsanity.com/paper/2103.01362/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 62 references that are in the map.
- First posted on arXiv 1 Mar 2021.

## Builds on (most cited first, 10 of 10)

- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [Data-driven reduced-order models via regularized operator inference for a single-injector combustion process](https://mlsanity.com/paper/2008.02862/) (2020)
- [Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms](https://mlsanity.com/paper/2002.09726/) (2020)
- [Data-driven Discovery of Closure Models](https://mlsanity.com/paper/1803.09318/) (2018)
- [Learning physics-based reduced-order models for a single-injector combustion process](https://mlsanity.com/paper/1908.03620/) (2019)
- [Sampling low-dimensional Markovian dynamics for pre-asymptotically recovering reduced models from data with operator inference](https://mlsanity.com/paper/1908.11233/) (2019)
- [Data-driven Structured Realization](https://mlsanity.com/paper/1611.02072/) (2016)
- [Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations](https://mlsanity.com/paper/2005.05890/) (2020)
- [Vector Fitting for Matrix-valued Rational Approximation](https://mlsanity.com/paper/1503.00411/) (2015)

## Built on by (most cited first, 5 of 5)

- [Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference](https://mlsanity.com/paper/2107.02597/) (2021)
- [Operator inference with roll outs for learning reduced models from scarce and low-quality data](https://mlsanity.com/paper/2212.01418/) (2022)
- [Active operator inference for learning low-dimensional dynamical-system models from noisy data](https://mlsanity.com/paper/2107.09256/) (2021)
- [Nested Operator Inference for Adaptive Data-Driven Learning of Reduced-order Models](https://mlsanity.com/paper/2508.11542/) (2025)
- [Dynamic Structure Estimation from Bandit Feedback using Nonvanishing Exponential Sums](https://mlsanity.com/paper/2206.00861/) (2022)

## Abstract (arXiv)

This work introduces a non-intrusive model reduction approach for learning reduced models from partially observed state trajectories of high-dimensional dynamical systems. The proposed approach compensates for the loss of information due to the partially observed states by constructing non-Markovian reduced models that make future-state predictions based on a history of reduced states, in contrast to traditional Markovian reduced models that rely on the current reduced state alone to predict the next state. The core contributions of this work are a data sampling scheme to sample partially observed states from high-dimensional dynamical systems and a formulation of a regression problem to fit the non-Markovian reduced terms to the sampled states. Under certain conditions, the proposed approach recovers from data the very same non-Markovian terms that one obtains with intrusive methods that require the governing equations and discrete operators of the high-dimensional dynamical system. Numerical results demonstrate that the proposed approach leads to non-Markovian reduced models that are predictive far beyond the training regime. Additionally, in the numerical experiments, the proposed approach learns non-Markovian reduced models from trajectories with only 20% observed state components that are about as accurate as traditional Markovian reduced models fitted to trajectories with 99% observed components.